0197804 58D4B Mangazeev p v Pankov m v Kulagina t e Kamartdinov m r Gidrod

5/23/2018 0197804 58D4B Mangazeev p v Pankov m v Kulagina t e Kamartdinov m r Gidrod

http:///reader/full/0197804-58d4b-mangazeev-p-v-pankov-m-v-kulagina-t-e-kamartdino

1

2003 YUKOS EP, All Rights Reserved

20032003


, ...

, .-..

, MSc Petroleum Engineering



2


1. Horne R. N. Modern Well Test Analysis // Petroway, Inc., PaloAlto. 1995.

2. Earlougher R.S. Advances in well test analysis // Monogr. Ser.Soc. Petrol. Eng. Dallas. Vol. 5. 1977.

3. Bourdarot G. Well Testing: Interpretation Methods. 1998.

4. .., .., , , 1992.

5. .. . .: ,1998.


6. 39-100-91

, --// , ,, ..,1991.

7. 153-39-007-96. : ..., 1996.

8. 153-39.0-109-01

,// ., 2002.



3


PANSystemPANSystem

Edinburgh Petroleum Services

http://www.e-petroleumservices.com

SaphirSaphir

Kappa Engineeringhttp://www.kappaeng.com


F.A.S.T.F.A.S.T.

Fekete Associates Inc.

http://www.fekete.com

PIEPIE

Well Test Solutions Ltd.

http://welltestsolutions.com



5



6

p

pws,

()

pwf

,

pi

pwf(t=0),

pe(re

)

pw

ps

pt=1

,

()

t = 1

0

*

mln

pwf

ln t/.

mlog

pwf

log t/.

c c -1

cw

-1

co

-1

cg -1

cf -1

P



7

ct -1

2/

D,

(3/)

-1

e .

,

.

ml

[,

].

CA Dietz .

tpDA

,

A , r2

.

D MBHMBH

.

S - .

Sa-,

.

Sd-

.

Sp-

.

Str - .

Sswp-,

.

D .

tD .

CD .

rD .

/3

L



8

r

rw

re

h

rinv

rs

hp

hs

rw,eff

xf

h d

l

b /A

2

g 2/

Q 3

V, Vw 3

Cs

()

3

/

Vu,

3/

B 3/

3

Bw 3/

3

B 3/

3

q 3/

qs 3/



9

q

3/

m

f

k

ks

km(

)

kf(

)

*/

T

tp

t,

tpe

Dte

tp

ti

tr,

tx

[,

]

So

Sw

Sg

Sn(t)

tps



10

Z

S-,

[t]M, [DP]M

[tD/CD]M,

[PD]M

P

-

(storativity ratio)

(transmissivity ratio)

,



11

1. 1.1.

1.2.

1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9.

2. -2.1. 2.2. , 2.3. 2.4. -

3.

3.1. 3.2. Cs 3.3. Cs , 3.4. 3.5. 3.6.

4. 4.1. 4.2. 4.3. 4.4. Gringarten4.5. 4.6. 4.7.

5. 5.1. 5.2. 5.3. 5.4. 5.5. c 5.6.



12

6.

6.1.

6.2. :

6.3. : MDH

6.4. 6.5.

7. 7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.7.

8. 8.1. 8.2. 8.3.

9.

9.1. 9.2. 9.3.

10.10.1. 10.2. 10.3. 10.4. 10.5.

10.6. :

11.11.1. 11.2. 11.3. 11.4.



13

12.12.1. 12.2. : 12.3. : 12.4. : -12.5. 12.6.

13.13.1. 13.2. 13.3. 13.4. 13.5. 13.6. 13.7. 13.8.

14.14.1. 14.2. 14.3. 14.4. 14.5. 14.6.

14.7.

14.8.

15.15.1. 15.2. RFT15.3. RFT15.4. 15.5.



15

:

- ,

- ,

.

-

.

-

.

:

:

-

.

- ,

.- .



17

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9



18

.

.

1.1 ,

, .. (, , ),, .

:

, , .. .

:

, ;

,;

,( , ..).



19

1.1

1.1

. , () (), ().

.

, ., (), ,.

, , . . , .



20

1.2

pwf

, t

q

0

0

, t

pws = pi

0

1.2

() t = 0() q.

: q = const; t = 0 , ..pi.

: , ,

; ,

.

, /. , , , , ,. , .



21

1.2

tp t

t

tp

pws

pwf(t=0)

1.2

() ,tpq,.

: q=const; tp

, .

: ;

, .

-.

() (q=0).



22

1.2

, t

q0

, t

q

, t

0

0

, t

-q

1.2

t = 0, .

.

, .

-:

, ;

.



23

1.2

1.2 :

; .

. /.

. () (,).

: (

);

; .

. .



24

1.2

1.2 ,

(, ) . , , .

() : ; ; ; PVT ; ; (-).

, .. .

, . , ,

, .. . .



25

1.2

RFT

MDT

1.2

, . (RFT - RepeatFormation Tester MDT Modular Dynamic Tester) . , , ,

. .

RFT MDT: ; ; .

, .

-: -; .



26

1.3

dx

dpk

A

Bqs

=

310644.8

A

q

x

dx

dpku

A

q==

,

1.3

:

= p gD ;D -; -.

. ()

:

q 3/ 3/ A c 2 2 u / / /2 p x k . 2 mD

k, : -( 100 %) ;

; .

dx

dk

dx

dDg

dx

dpku

A

q

=

==



27

1.3 1

,

, k:

a) t = 20C ( = 1 ) = 0.68. Q = 2.88 x 10-3 3/,

b) = 1.1 , ) Q = 10.468 x 10-3 3/.

: L= 0.05 , A = 5 10-4 2. b.



28

1.3

dr

dpk

rh

qur

==

2

rwre

r

ur

q

qq

pe

h

w

ewe

r

r

kh

qpp ln41.18=

ur

pe

rwr

re

1.3

.

,hk, . (re) pe, pw. (), , (). (). hq (),

,.

q, pe-pw :

., q - .

w

ewe

r

r

kh

qpp ln41.18

=



29

1.3 2

10 100 -

, ,

k = 0,5 , h = l0 ,

w = 80 a, rw = 12.4

, o = 4 , q = 229.885 3/.



30

1.4

constTp

V

Vc

=

=1

1=++

+++=

gwo

fggwwoot

SSS

ccScScSc

V [3] ; [/3] ;cw [

-1] ; B ;

co [-1] ; S ;

cg [-1] ; Sw ;

cf [-1] ; Sg .

T

o

oT

o

oT

o

o

op

B

Bpp

V

Vc

=

=

=

111

1.4 c

. (1/)

co, .. . .

cf, .. .

ct, , , ().

, =136 .: o = 2.2 x 10-4 1/ cw = 4.4 10

-5 1/

cg = 7.3 10-3

1/ 1/ ,

. , t cg



31

1.5

p ;

r ;t ;

= k / ct ;

k ;

;

;

ct

t

p

k

c

r

p

rr

p t

=

+ 1

2

2

dr

dpkur =

( )

tr

ur

r

r

=

1

p

co

=

1

1.5

, , , .

: (); ; .

, : ; , (kx = ky = kz); ; q k (); ; -; ;

.



32

1.5

1

2

3

0.4 1.0 1.6

Ei(x)

-Ei(-x)

0

x

( ) duuexEi

x

u =

1.5

,. , , ,

,

.

:

pi; (); rw



33

1.6

t

invc

ktr

037.0=

1.6

,. , -rinv, , .

rinv, ,.

, (): Jones: rinv,

1% .

Poettmann: rinv, 1% .

J.LeeMuskat: rinv, .



34

1.6 -

rinv :

a) k= 100 = 0.25 ct = 1.47 10

-4 -1

= 0.8 t = 1

b)

k= 0.010 = 0.05 ct = 2.939 10-3 -1

= 0.05 t = 168

103 104 t=105

t

invc

ktr

037.0=



35

1.7

1.7

. r.

, , rw., ,, .

rw. , .



36

1.7

,

,

1.7

.

,:

;

.



37

1.7

1.7



38

1.7

1.7

, , .

:

-, ;

-,.

, -.



39

1.8

,

t

rw r re

p

1.8

., .. , .

/, .., /.

.. .

,.

constPt

Pe==

0

( ) ( )trfptrft

P,, ==

errrPconst

tP ===

,0



40

1.8

MTR LTR

pi

pwf

0

1.8

:

. , ()

MTR c ()

LTR ()



41

1.9

. 1q1

. 2

q2

r1

r2

. 3

2,31,33ppp +=

1.9

, , ,, ., .

.

(), , .: ; ; .

:

() (), .. -

a x, b y, a + b x + y



42

1.9

q

0

q

0

0

-q

+

=

t

t

tp

tp

pw

+

=

pw

pw

pi

pi

pi

p(t)

p(t)

tp

t

tp

1.9

, .

: tq, .

,: t = 0q1 = q, tp c q2 = -q ()

, q1q2



43

1.9

q q

q -q

1.9 ,

, . . , .

,:

, , , ,

, ,

, , , ,,

. , ,

,



44

1

1. ?

2.

?

3. ?

4. ,

?

1

5. cg?

6. ?

7. :

a. , ;

b. RFT/MDT;

c. ;

d. .



45

--

2.1 -

2.2

2.3 2.4 -



46

2.1 -

S < 0

S ~ -3

S ~ -4

S ~ -5.5

S > 0

S ~ 1-2

S ~ 5

S > 10

ps

pw

pwf

"

t

rw

ps

pwf

pwks > k

t

(

)

()

kh

q

pS s

41.18

=

2.1 -

: . , . : :

;

; , , CaCO3 BaSO4; ; ; (

). :

-;

; .

, , : -; .



47

2.1 -

- - Van Everdingen Hurst

-

pw

pwf

k

ps

ps -()

S -

kh

q

pS s

41.18

=

pwf = pw + ps

2.1 -

, . , (-) , .

- : p

wf

= pw

+ ps

, - ps qsB, - (rs, ks). pw (k = const) .

, -, - (Van Everdingen Hurst) :

- , - ( ) , .

, -1.

kh

q

pS s

41.18

=



48

2.1 -

-

ks

rs

pwf

ps

kspe

rsrw

ps

(

k)

ks

pw

pwf

s

s w

rk

S 1 lnk r

=

2.1 -

-., - rs ks; : k h. --rs



49

2.2

, ..

2.2

, . , -, , .

, ,

, -. kr, kz.

BronsMarting, -.



50

2.2

h =

hp =

hs =

h h hhs

hshs

h

hp

hp

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

30

25

20

15

10

5

S

b

hD= 10000

20

100

5

1000

b =

Brons

Marting

-

z

r

w

sD

k

k

r

hh =

2.2

: b

hp/h hs/rw

bhs/rw, , , .

BronsMarting, .. kz=kr. kz < kr.

BilhartzRamey, , ,BronsMarting, hs/rw:

-, BronsMarting, .

z

r

w

sD

k

k

r

hh =



51

2.2

(S=0)

rD

Sp

SdSa

pD

-

-

ptr

pda Sb

SSSS +=+=

2.2

,,

;

- (, , )

:Sa = Sd + SpJones Watts

Sp =-,Sd =-(

). -

, - : Sd = Str/ b, Str= -;

b = .

-:Sa = Str/ b + Sp

, - Str, , () ( ).



52

2.2

-

30.48 (h = 91.44).

- Sa = 20.

, 30.48 , .

:

kr = 180 rw= 0.0762 o = 1.0

kz = 5 re = 304.8 Bo = 1.2



53

2.3

h

0

-2

-4

-6

102 103 104

Sswp

hrw

30o

45o

60o

= 75o

15

Cinco &

Miller

2.3

, .

() Cinco, Miller Ramey. - Sswp.

Sswp h / rw. :

- , ..

.

. , - Sp Sswp-.

2.06 1.865

swp

w

w

hS log

41 56 100r

h 0 75 40

r

=

< < >



54

2.4

-

r r ew eff wS

=

pwf

pe

ps

rerwreff

h

re

xf

xf

xf < < re

xf =

2.4 -

. , ..

re >> xf, -Spr.

rw eff. rw eff, ,

, () .- :

(Prats) , - :

s

weffw err =

f

w

effw

wpr

f

effw

x

r

r

rS

xr

2lnln

2

,

,

==

=



55

2.4

-

-

10600

10500

10400

10300

11100

11000

10900

10800

10700

10200

10600

10100

1050010000

10400

10300

9900

9800

9700

960010100

10200

100009500

10600

10500

10400

10200

10000

9900

9800

9700

9600

9500

10100

10300

10200

10300

10400

10500

10600

10700

10800

10900

11000

11100

11200

11300

10600

10500

10400

10300

10200

11200

11300

11400

11500

11100

10100

10000

9900

9800

9700

9600

11000

10900

10800

10700

10600

10500

10400

10300

9500

10600

10500

10400

10200

10000

9900

9800

9700

9600

9500

10100

10300

10100

10200

10300

10400

10500

10600

10700

10800

10900

11000

11100

11200

11300

Core DST

2.4 -

- : ,

: S > 0;

(): S < 0.

-, .. , , .

: . -,, .

-(,..) (k/). ,-, ,.

, .



56

2

1. ?

2. - ?

3. kz?

4. , , -?

5. 6 , -?



57

3.1

3.2 Cs

3.3 Cs,

3.4

3.5

3.6



58

3.1

3.1

-,

()

. :

/



59

3.1

q

t

0

s

VC

P

=

3.1

, ,

Cs, : Cs=-V / P

Cs

[3/]



60

3.2 Cs

wV cV P =

sVCP

=

Vw [3]

c [-1]

s w

t = 0() q

0 q.

, Pi

,

:

Vw[3]; [-1]

: 0,010,1 3/

wV cV P =

s wC c V=

3.2 Cs



61

3.3 Cs,

uV V h =

s

VC

P

=

us

Vu ,

[2]

h []

1P 101325 g h =

, , :

Vu, [2]

h []

h:

- , [/3]

: 0,1 1 3/

:,

uV V h =

1P 101325 g h

=

uVC 101325g

=

3.3 Cs,



62

3.4

P , []

q , [3

/]B

t , []

Cs

, [3/]

s

qBtP 24 C =

, ~ 0

:

P , []; q ,[3/]; B ; t , []; Cs, [3/]

s

qBtP

24 C

=

3.4



63

, Cs,

s

qBtP

24 C =

t

P

0

n

s

3.4

, ,(~ 0), :

n :

-: /

s

qBt

P 24 C =

s

qBC

24 n=

3.4



64

3.5

sf

24C dPq q

B dt= +

qf , [3/]

,

:

qf , [3/]; q

, [3/]; B ; t ,[] ; P , []; Cs, [3/]

sf

24C dPq q

B dt= +

3.5



65

(Ramey):

(Chen) (Brigham):

t - []; Cs -

[3/] ; k -

[] ; h - [] ; -

[]

3.6

( ) s26531 1547 S Ct

kh

+ =

s22109 C exp(0.14 S)tkh

=

:

(Ramey):

(Chen and Brigham)

k , []; h , []; , []; s -

( ) s26531 1547 S Ctkh

+ =

s22109 C exp(0.14 S)tkh

=

3.6



66

:

10

1

0.1

p

0.01 0.1 1 10t

100

1,5 log-

3.6

:

,

, ()1,5

3.6



67

1 .

:

240

245

250

255

260

265

270

-15 0 15 30 45 60 75,

,

-20

0

20

40

60

80

100

120

,

3/

1

3

/110q

3/31B



68

3

1.

2. ,

3. ( , )

4. : ; ;

5. ,



69

4.1

4.2

4.3

4.4 Gringarten

4.5

4.6

4.7



70

4.1

70-

P(t) -

Agarwal

McKinley

EarlougherKerschGringarten

70-

, . :

Agarwal

McKinley

EarlougherKersch

Gringarten

--. .

Gringarten.

4.1



71

4.2

D 2

t w

0.00036kt t

c r=

D

khP P

18.41qB=

D s2

t w

0.159C Ch c r

=

S

:

-

k []; h []; q , [3/]; B ; []; t , [];P , []; ; ct[1/]; rw []; Cs ,

[3

/] -

, ,

D 2

t w

0.00036kt t

c r=

D

khP P

18.41qB=

D s2

t w

0.159C Ch c r=

S

4.2



72

DD

D

tP

C=

D D

1P ln t 0.81 2S

2= + +

4.3

:

:

:

.. ,

DD

D

tP

C=

( )D D1P ln t 0.81 2S2= + +

4.3



73

PD

tD/CD

CDexp(2S)

4.4 Gringarten

, : , -:

Gringarten PD:

, GringartenPD tD/CD ;CDexp(2S)

: , CDexp(2S)>1,

CDexp(2S)



74

D

D S

t 0.00226kh

tC C=

D

khP P

18.41qB=

D

khlog P log P log

18.4qB

= +

D

D S

t 0.00226khlog log t log

C C

= +

4.4 Gringarten

:

, ,

.

( )Dkh

log P log P log18.41qB

= +

D

khP P

18.41qB

=

D

D S

t 0.00226kht

C C=

D

D S

t 0.00226khlog log t log

C C

= +

4.4 Gringarten



75

4.5

10

1

0.1

P

0.01 0.1 1 10t

100

:

P(t) . !

4.5



76

100

PD

0.1 104

0.01 0.1 1 10t

P

10

1

100

1 100 10000.1

tD/CD

M

4.5

, , . !

CDexp(2S) ( )

: ([t]M, [P]M) ([tD /CD] M, [PD ]M)

4.5



77

MS

D D M

t0.00226khC

t C

=

D M

M

P

kh 18.41qB P=

D

D

C exp 2S1S ln

2 C

=

4.5

, kh:

, CS:

CDexp(2S)-:

[ ]

[ ]M

S

D D M

t0.00226khC

t C

=

[ ]

[ ]D M

M

Pkh 18.41qB

P=

( )D

D

C exp 2S1S ln

2 C

=

D s2

t w

0.159C C

h c r=

4.5



78

4.6

PP

t, t

i wfP P P = ( ) ws wsP P P t 0 = =

t



79

10

1

0.1

P

0.01 0.1 1 10te

100

e

p

tt

t1

t

=

+

4.6

, te,

. : P(te)

, , , , tp

e

p

tt

t1

t

=

+

4.6



80

4.7

-

:

-(-)

,

():

.

4.7



81

1 .

: . , -

240

245

250

255

260

265

270

-15 0 15 30 45 60 75,

,

-20

0

20

40

60

80

100

120

,

3/

1

245.4p(t = 0)

3

/110q

48tp

1/2.20E-04ct

1

3/31B 0.08rw

80h

0.2



82

1.

2. ?

3. (, , ) ?

4. ?

5. ?

6. C

Dexp(2S)?

7. . ?

8.

4



83

5.1

5.2

5.3

5.4 5.5 c

5.6



84

5.1

( )P

Pln t

=

ln t

lnP

ln t

P

,. ,

80- .

1983 (Bourdet):

, P , P(t) (. )

. ..

( )P

Pln t

=

5.1



85

tD/CD

0.1 1 10 100 1000

PDPD

1

10

0.1

1

1010010001e4

1e61e81e10

1e15

1e20

1e30

1e401e50

0.1

1

CDexp(2S)

0.5

5.2

:

:

, ,0,5

DD

D

D

PP 0.5

tln

C

= =

( )D D1

P ln t 0.81 2S2

= + +

( )DD DD

t1P ln 0.81 ln C exp 2S

2 C

= + +

5.2



86

tD/CD

0.1 1 10 100 1000

PDPD

1

10

0.1

1

1010010001e4

1e61e81e10

1e15

1e20

1e30

1e401e50

0.1

1

CDexp(2S)

= 1

5.2

:

:

( )P PP t

ln t t = =

DD

D

tP

C=

DD

D

tP

C=

5.2



87

D

D

D

PP a

tln

C

= =

DD

D

tP a ln b

C

= +

, .. :

5.2

, , :

(, .. ):

:

,

D

D

D

PP a

tln

C

= =

DD

D

tP a ln b

C

= +

5.2



88

n

DD

D

tP a n

C

=

n

DD

D

tP a b

C

= +

( )DDD

tlog P n log log a n

C

= +

, , , ,

:

5.2

(, , ,, ):

:

, n

n 1 n

D D D DD

D D D DD

D

t t t tPP a n a n

C C C Ct

C

= = =

n

DD

D

tP a b

C

= +

( )DDD

tlog P n log log a n

C

= +

5.2



89

!

5.2

, , , ()

5.2



90

5.3

( )( )

( )i iP

P t tln t

=

( ) i i 1ii i 1

P PP t

ln t ln t

( ) i 1 i 1ii 1 i 1

P PP t

ln t ln t

+

+

( ) i 1 iii 1 i

P PP t

ln t ln t

+

+

:

Pi ti

:

5.3

( )( )

( )i iP

P t tln t

=

( ) i i 1ii i 1

P PP t

ln t ln t

( ) i 1 iii 1 i

P PP t

ln t ln t

+

+

( ) i 1 i 1ii 1 i 1

P PP t

ln t ln t

+

+



91

5.3

( ) i i 1ii i 1

P PP t

ln t ln t

( ) i 1 i 1ii 1 i 1

P PP t

ln t ln t

+

+

( ) i 1 iii 1 i

P PP t

ln t ln t

+

+

ln t

P

, ln tiPiti:

5.3

-9.25278.490.74

1.38-9.2512.00278.860.70

7.5012.003.00278.380.66

-0.633.00-4.25278.260.62

0.87-4.256.00278.430.58

4.256.002.50278.190.54

5.002.507.50278.090.50

6.257.505.00277.790.46

3.885.002.75277.590.42

5.002.757.25277.480.38

13.887.2520.50277.190.34

8.3820.50-3.75276.370.30

12.25-3.7528.25276.520.26

23.1228.2518.00275.390.22

33.0018.0048.00274.670.18

48.00272.750.14

Piln ti

, ,



92

5.3

l1 l2

m1

m2

1 2 2 1i

1 2

m l m lm

l l

+=

+

, ti, [ln ti -; ln ti + ]

:

, [ln ti -; ln ti + ], . ti

(ln ti) (ln ti) , m1. (ln ti) (ln ti +) , m2. ti m1 m2. ,

, ti

, [ln ti -; ln ti + ]. m1 m2( . ):

5.3

1 2 2 1i

1 2

m l m lm

l l

+=

+



93

5.3

2< 0.35

0.01 0.1 1 10

b

2

2



94

5.4

:

,

5.4



95

10

0.01 0.1 1 10 100

PP

t

100

5.5 c

, 4:

.

!

5.5 c



96

0.10.1

1

10

100

1 10 100 103

104

PDPD

tD/CD

0.5

PP

t

100

10

0.01 0.1 1 10 1001

5.5 c

. :

~ : ,

~ : , 1, , 1

CDexp(2S)

( ) : ([t]M, [P]M) ([tD /CD] M, [PD ]M)

, 4

5.5 c



97

5.6

t

P

t

P

, :

, 0,5.

:

- :

, t

kh0.5 P18.41qB

=

s

qBP t

24C =

2

t w

0.00036kt9.205qBP ln 0.81 2Skh c r

= + +

5.6



98

s

tqBC

24 P=

0.5kh 18.41qBP=

2

t w

P 0.00036kt1S ln 0.81

2 P c r

=

5.6

, :

0.5kh 18.41qB

P=

s

tqBC

24 P=

2

t w

P 0.00036kt1S ln 0.81

2 P c r

=

5.6



99

1 ( ) .

: . , -

240

245

250

255

260

265

270

-15 0 15 30 45 60 75

,

,

-20

0

20

40

60

80

100

120

,

3/

1

245.4p(t = 0)

3/110q

48tp

1/2.20E-04ct

1

3/31B

0.08rw

80h

0.2



100

1. , , ?

2. ?

3. ?

4. ?

5



101

--

6.1

6.2

:

6.3: MDH

6.4

6.5



102

6

ln r

tp(r,t)

pi

q

Pwf

Pi

(

)

,

()

()

6

,. , , . ,

, . ,

, .

-.

: (, ), ,-, ():

=

D

DD

t

rEip

42

1 2



103

6.1

104 105 106 107 108 10910

1

10-1

10-2

10-1 1 10 102 103 104

pD

tD / rD2

(S=0)

6.1

Ei(x), . , , x < 0.01, :

= 1.781.

.. :

(rD = r / rw = 1).

:

rD = 1,-:

+

=

= 80907.0ln

2

1

4ln

2

12

2

D

D

D

DD

r

t

t

rp

( )

+

= 80907.0

00036.0ln205.9,

2rc

kt

kh

Bqptrp

t

si

25

2>

rc

kt

t

( ) ( ) ( )xxEixEi == ln

01.04

2

wtrc

kt



104

6.1

0

pt= 1

ln t

: ln t = 0 t = 1

,

pwf

kh

Bqm s205.9

ln

=

6.1

, , :

, rD = 1:

:

pi[];t [].

pwfln t () :

( = arctg mln );

,

() t = 1,.. ln t = 0.

[ ]Stp DwfD 280907.0ln2

1++=

( )

++= S

rc

kt

kh

Bqptp

wt

siwf 212034.7lnln205.9 2

kh

Bqm s205.9ln =

1ln ln =+= twf ptmp

+== S

rc

kmpp

wt

it 212034.7ln 2ln1



105

6.1

( )

++= S

rc

kt

kh

Bqptp

wt

siwf 212034.7lnln205.9 2

hm

Bqk s

ln

205.9=

+

= = 12034.7ln

2

12

ln

1

wt

ti

rc

k

m

ppS

( )

++= S

rckt

khBqptp

wt

siwf 86859.009232.3loglog195.21 2

+

= = 09232.3log1513.1

2

log

1

wt

ti

rc

k

m

ppS

hm

Bqk s

log

195.21=

1log1ln logln == +=+= twftwf ptmpptmp

pwf

ln t

pwf

log t

6.1

ln t.

pwfln t;

., , .

log t.

, pwflog t, , .

: , . , pt=1.

( )

++= S

rc

kt

kh

Bqptp

wt

siwf 212034.7lnln205.9 2

( )

++= S

rc

kt

kh

Bqptp

wt

siwf 86859.009232.3loglog195.21 2



106

1

245

247

249

251

253

255

257

259

261

263

265

0 20 40 60 80 100 120

t,

,

0

20

40

60

80

100

120

q,

3/

q

1

. , .

:

= 0.2;

h = 80 ; rw = 0.08 ;

Bo = 1;

= 1 ;

t = 2.2 x 10-4 1/;

q = 110 3/;

pi = 265.



107

6.2

:

pwf pws

pwfextp

pBU

t

pi

pwf(t=0)

pw

( )

+=+=

t

tt

kh

Bqtpttpp

psDDpDDBU ln205.9

6.2 tp q,

t, , , .. :

q (tp + t) (0-q) t

:

: ; , tp,

.

, :

, pwsln [(tp + t)/t]

p* = pi.

( ) ( ) ( ){ }DwDDpDwDsiws tpttpkhBq

ptp += 41.18

( )

+=

t

tt

kh

Bqptp

psiws ln205.9

kh

Bqm s

205.9ln =



108

6.2

pws

lntp + t

t

p*

ln 1

,

kh

Bqm s205.9ln=

pws(t=1)

ln (tp + 1)

6.2 kh m

( )

pws ln((tp + t) / t)

pws log((tp + t) / t)

- , :

1 t = 1:

pwf(tp) = pws(t = 0):

- pws ln((tp + t) / t)pws log((tp + t) / t) :

( ) ( )

+

++

== 09232.3log

1log

11513.1

2

log wtp

ppwfws

rc

k

t

t

m

tptpS

( ) ( )1ln205.91 +== pwsi tkh

qBtpp

log

ln

195.21

205.9

m

Bqkh

m

Bqkh

s

s

=

=

( )

++= S

rc

kt

kh

qBtpp

wt

ppwfi 212034.7lnln205.9 2

( ) ( )

+

++

== 12034.7ln

1ln

1

21 2

ln wtp

ppwfws

rck

t

t

m

tptpS



109

6.2

pws

=

lntp + t

t

pws

lin

0

p*

p

khBqs205.9

6.2

(tp + t) / t = 1, t. p* = pi , . , , .

.

, tp, p

* . p* , MBH (Matthews-Brons-Hazerbroek).



110

2

245

247

249

251

253

255

257

259

261

263

265

0 10 20 30 40 50 60 70 80

t,

pws,

tp = 48

q = 110 3/

p(t=0) = 245.4

2

1., qtp, 72 pws. , .

:

= 0.2;

h = 80 ;

rw = 0.08 ;

Bo = 1;

= 1 ;

t = 2.2 x 10-4 1/;

tp = 48 ; q = 110 3/;

p(t=0) = 245.4 .



111

6.3

:

MDH

pw

pwf(tp)

0

pwf

pDDpMDH

t

t

pws

p

tp

pi

6.3 MDH

ln[(tp + t) / t].

, , tp >> t:

.. lnt. , .

p pMDH tp >> t, .:

;

.

( ) ( )pswsi ttkh

Bqtpp lnln205.9 =



112

6.3 MDH

( )

+

== 12034.7ln12

1 2ln wt

pwfws

rck

mtptpS

ln

205.9

m

Bqkh s=

( )

+

== 09232.3log

11513.1

2

log wt

pwfws

rc

k

m

tptpS

log

195.21m

Bqkh s=

0

2

4

6

8

10

12

14

16

18

20

-3 -2 -1 0 1 2

6.3 MDH

MDHpMDHlnt (pMDHlog t )mln (mlog).

,:

p*; tp >> t



113

6.4

t

q

peq

Qt

q

q

tp

6.4

. , tpq().

tpe:

Q .

tpe, , ., , (, ), , .

,

, ().

pe q

Q

t 24



114

331

332

333334

335

336

337

338

339

340

341

0 100 200 300 400 500

0

5

10

15

20

25

30

35

t,

p,

q,3/

,

.

, .

:

= 0.36;

h = 5.3 ;

rw = 0.08 ;

Bo = 1.2;

=0.5;

t = 1.2 x 10-4 1/.



115

0

5

10

15

20

25

30

35

0 100 200 300 400 500

,3/

t,

tpe =350

q1

q2

q3

q4

q5

:

t = 0..100 , q1 = 31.8 3/

t = 100..200 , q2 = 0 3/

t = 200..300 , q3 = 7.95 3/

t = 300..400 , q4 = 15.9 3/

t = 400..500 , q5 = 0 3/

q4 = 15.9

3/. ,:

tpe:

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ]3333 9.23124

1009.15

24

10095.7

24

1008.31

Q =++=

[ ][ ]

q

Qt

pe 350/9.15

9.2312424

3

3

==



116

y = -0.4034x + 340.26

336.0

336.5

337.0

337.5

338.0

338.5

339.0

339.5

340.0

0123456

log [(tpe + t)/ t]

pws

pws(t=1)

,:

mlog = 0.4034/.

, () t = 1,.. log (tp + 1) = 2.55, pws(t=1) = 339.2 .

, mlogpws(t=1).



117

hm

Bqk s 94195.21

log

==

( )6.209232.3log

1log

11513.1

2

log

=

+

++

==

wtp

ppwfws

rc

k

t

t

m

tptpS

336.0

336.5

337.0

337.5

338.0

338.5

339.0

339.5

340.0

0123456

m :

pws(t=1) -:

, .. , .

[ ][ ] [ ]

hm

Bqk 94

3.5.4034.0

5.02.19.15195.21195.21

3

log

4 =

==

( ) ( )

[ ] [ ][ ]

[ ][ ]

[ ][ ] [ ] [ ]

6.209232.308.01102.15.036.0

94log

350

1350log

./4034.0

3362.3391513.1

09232.3log1

log1

1513.1

224

2

log

=

+

+

+

=

=

+

++

==

rc

k

t

t

m

tptpS

wtp

ppwfws



118

6.5

6.5

, :

, , tN, :

, ,

(tp + t) / t

( ) ( ) ( )11

141.18 =

= iNDN

i

iiwfi ttpqqkh

Btpp

( ) ( )

( )

( )

+

+

=

=

Src

tkqq

ttt

ttqq

kh

Btptp

wt

NN

iN

iNN

i

ii

Nwfws

212034.7ln

ln

205.9

21

1

1

1

( ) ( ) ( ) tqqttt

ttqqtSn NN

iN

iNN

i

ii +

=

= lnln 1

1

1

1



119

6.5

p

*

p

Sn(t)Sn(t=1)

p1

khBm 205.9ln =,

6.5

pws(t) Sn(t), m, , . m , .. , , .

m kh:

- S p1.

ln

205.9m

Bkh=

( )

( )

+

=

12034.7ln2

12

ln1

1

wtNN

Nwf

rc

k

mqq

tppS



120

()

,3/

t,

0

5

10

15

20

25

30

35

0 100 200 300 400 500

q1

q2

q3 q4

q5

t1 t2 t3 t4 t5

t

( ) ( ) ( ) ( )tqqttt

ttqqtSn

i

i

i

ii +

=

= lnln 45

5

54

1

1

()

.

qiti, .

t = 1, pws(t=1).

( ) ( )

( ) ( ) ( ) tqqttt

ttqq

ttt

ttqq

ttt

ttqq

ttt

ttqtSn

+

+

+

+

++

+

+

=

lnlnln

lnln

45

45

4534

35

3523

25

2512

15

151

( ) ( )

( ) ( ) 1ln1ln

1ln

1ln1

45

4534

35

3523

25

2512

15

151

+

++

+

++

+

+

==

tt

ttqqtt

ttqq

tt

ttqq

tt

ttqtSn



121

()

y = 0.0114x + 339.23

336.0

336.5

337.0

337.5

338.0

338.5

339.0

339.5

340.0

-110 -60 -10 40Sn(t)

pws

pws(t=1)

Sn(t=1)

pws(t) Sn(t)

()

pws(t) Sn(t) , , . ,:

mln = 0.0114/3//.

, ()

t = 1,.. Sn(t = 1 ) = -0.09, pws(t=1) = 339.23 . ,

mlnpws(t=1).



122

()

336.0

336.5

337.0

337.5

338.0

338.5

339.0

339.5

340.0

-110 -60 -10 40

hm

Bk 5.90205.9

ln

==

( )

( )2.212034.7ln

2

12

ln1

1 =

+

=

wtNN

Nwf

rc

k

mqq

tppS

()

mpws(t)Sn(t) :

pws(t=1) -:

, , , .

[ ] [ ]

hm

Bk 5.90

3.5.0114.0

5.02.1205.9205.9

3

ln

=

==

( ) ( )

( )

[ ] [ ]( )[ ] [ ]

[ ][ ] [ ] [ ]

2.212034.708.01102.15.036.0

5.90ln

./0114.009.15

3362.339

2

1

12034.7ln1

2

1

224

33

2

ln54

5

=

+

=

=

+

==

rc

k

mqq

tptpS

wt

wfws



123

6

1. /:

a) ;

b) ;c) .

2. , :

a) ;

b) p1hr;

c) ;

d) .

6

3. - , :

a) ;

b) p1hr;

c) ;

d) .

4. MDH ?

5. ?

6. ?



125

7.1

7.2

7.3

7.47.5

7.6

7.7



126

7.1

, , . -,

,

:

, ,

7.1



127

7.2

. :

7.2



128

( ) ( )D D D D D D DP P t , r 1, S P t , 2r , 0= = +

d d d

7.2

, d, :

,

, , 2d

D

w

dr

r=

( ) ( )D D D D D D DP P t , r 1, S P t , 2r , 0= = +

7.2



129

D D

1P ln t 0.81 2S

2= + +

( )2

D

D D i

D

2r1P ln t 0.81 2S E2 4t

= + +

( )D D DP ln t 0.81 S ln 2r = + +

P

ln t

m

2m

7.2

, :

, , Ei-. ,(m):

, ,. ,

(t), Ei-:

, :

,

( )2

D D2r 4t

( )2

D D i D D

1P ln t 0.81 2S E 2r 4t

2 = + +

[ ]D DP 0.5 ln t 0.81 2S= + +

( )2

D D2r 4t 0.01>l, .:

, :

7.4

( )

( ) ( ) ( )( )

D D D

D D

4 t 2 r ln 1 exp 2 r

1ln 1 exp 2 r cos 2 e exp 1 exp 4 r

2

= + +

+ +

l l l

l l

D D4 t 2 r = l l

x

t

kta 0.01

c=



139

0,1

1

10

100

0,001 0,01 0,1 1 10 100 1000 10000

log P

log P

log t0,1

1

10

100

0,001 0,01 0,1 1 10 100 1000 10000

log P

log P

log t

7.4

. (P t1/2) 2

,

, ( > ).

7.4



140

7.5

,

:

7.5



141

ln t

P m

2m /

2m

1

2

m2

m = -

7.5

, , , , m1=m,. , m2=2m/, ,

, 2m,

:

,

1

2

m2

m =

7.5



142

log t

log P

log P

mln

2mln

2mln/

7.5

-,

, 2mln/

, 2mln/, 2mln

7.5



143

log t

logP

log P

1

90

45

1

7.5

, . ,

mln2mln/ - , 0,5

7.5



144

7.6

d

, :

,

7.6



145

7.6

d 2dq q -q

d 2dq q -q

d 2dq q -q

d 2dq q -q

d 2dq q -q

d 2dq q -q

d 2dq q -q

d 2dq q -q

( ) ( )D D D D D D DP P t , r 1, S P t , 2r , 0= =

2d

, :

,

7.6

( ) ( )D D D D D D DP P t , r 1, S P t , 2r , 0= =



146

D1P ln t 0.81 2S2

= + +

( )2DD i

D

2r1P ln t 0.81 2S E

2 4t

= + + +

( )D DP S ln 2r = +

7.6

, (, ):

, , :

()

:

:

:

( )2

D D2r 4t

[ ]DP 0.5 ln t 0.81 2S= + +

( )2

D D2r 4t 0.01CA2. , tptSSS

3. P*

4. tpDA, tptSSS:

5. PMBH

6. :

p /SSS

pDA

t

0.00036ktt

c A=

log* *

ln DMBH DMBH

mP P m P P P

2.303= =

7.7

( ) tSSS DA SSSc A

t t0.00036k

=



161

*

ln A pDAP P m ln C t=

tpDA

PD

MBH=(P*-P)/mln

DMBH A pDAP ln C t=pDA D SSSt t>

, MBH

7.7

, , PD MBHlog tpDA

:

, , :

( )DMBH A pDAP ln C t=

( )* ln A pDAP P m ln C t=

7.7



162

1 .

:

263.4

263.6

263.8

264

264.2

264.4

264.6

264.8

265

0 20 40 60 80 100 120 140 160 180 200

,

,

0

50

100

150

,

3/

3/110q

265Pi

1/2.20E-04ct

1

3/31B

0.08rw

80h

0.2

1



163

2 .

:

263

263.2

263.4

263.6

263.8

264

264.2

264.4

264.6

264.8

265

0 50 100 150 200 250 300 350 400 450 500

,

,

0

50

100

150

,

3/

265Pi

3/110q

1/2.20E-04ct

1

3/31B

0.08rw

80h

0.2

k - S 1

2



164

3 .

:

262

262.5

263

263.5

264

264.5

265

0 20 40 60 80 100 120 140 160 180 200,

,

0

50

100

150

,

3/

265Pi

3/110q

1/2.20E-04ct

1

3/31B

0.08rw

80h

0.2

3

k - S 1



165

160

170

180

190

200

210

220

230

240

250

260

355 360 365 370 375 380 385,

,

-20

0

20

40

60

80

100

120

140

160

,

3/

150 3/ 360 24 . :

, . A=640 0002; CA=10.837

:

4

265Pi

7k

360tp 2640 000A

10.837CA

3/150q

1/5.0E-05ct

1

3/31B

0.08rw

40h

0.2

4



166

1. ?

2.

?

3. , ,

4. , , ?

5. ?

6. ?

7

7. ? ?

8. ? ?

9. , ? ?

10. ??

11. P*MBH?

7



167

8.1

8.2

8.3



168

8.1

( Matthews Russell)

8.1

, () . ,

.



169

8.1

, km, m, cm

, kf, f, cf

PqB

hkP

f

D =41.18

mmff

ff

cc

c

+

=

( ) 200036.0

wmmff

f

Drcc

tkt

+

=

2 :

- (storativity

ratio, ): 0.01 - 0.1, 0.1 0.2

(transmissivity ratio, ): 10-3 10-7

2

w

f

m rk

k=

8.1

, .

. - (storativity ratio, ), . , - . 0.01 - 0.1, 0.1 0.2.

(transmissivity ratio, ), . 10-3 10-7. . .

.



170

8.1

m

2

m

2

f

2w

f

m

8.1

Warren Root , . , .

(xm

)



171

8.1

8.1

. , - . , . ,

, , . . , , .

, - . ,

.



172

8.1

+=

111

min

12

1

D

DD

t

pt

94903.0

log01765.0

log min

+

= D

DD

t

pt

min =Dt

8.1

.

- , , .

, .

w



173

8.1

1lnmin =Dt

-

tDmin

mP

=10

+

++

= = 09232.31

log

1

log1513.12

01

p

p

wt

thr

t

t

rc

k

m

PPS

2

00036.0

wt

f

Drc

tk

t

=

m

qBkh

195.21=

8.1

. . , . -

. , , .

, , . .



174

8.1

8.1



175

8.1

, ,. 95 3/(600 bbl/d) 300 .

600 . , -

(-):

1.2

20 (65.6 ft)

1.5

0.15

1.1*10-4 -1 (7.5*10-6psi-1)

.

8.1

228.43357.4610.51547

226.13323.7460.39288

223.73288.5790.29947

221.23251.7040.22827

218.63212.6960.17398

215.73171.0910.1326

212.73126.4490.10107

209.43078.4870.07706

205.93027.1680.05872

202.32973.2950.04477

198.52917.7920.03412

194.72862.2730.026

191.12808.6090.01984

187.62757.8230.01511

184.52711.4870.01151

181.72670.7460.00879

179.22634.8550.00668

177.22604.7220.0051

175.52579.3960.00388

174.12558.9540.00296

172.92542.320.00226

172.02528.650.00171

171.32518.3630.00131

170.82510.1910.00101

168.92482.4830

, , psi,

261.33840.826457.32922

260.43827.165348.58411

259.33811.423265.69678

258.13793.763202.51859

256.83774.634154.36313

255.43754.764117.6582

254.13735.04289.68106

252.83716.32768.35641

251.73699.27252.10242

250.63684.22139.71332

249.73671.13730.27017

249.03659.57723.07242

248.23648.75117.58618

247.53637.69613.40448

246.63625.49210.21713

245.73611.437.78766

244.63595.0925.93588

243.33576.3254.52444

241.83555.1853.44861

240.33531.8452.62857

238.53506.542.00354

236.73479.5081.52713

234.83450.9561.164

232.73421.0460.88724

230.63389.8730.67627



176

8.2

k

k

8.2

2- , , . . .



177

8.2

,

k1, h1

k2, h2

PqB

hkhkPD

+=

41.18

2211

2211

11

hkhk

hk

+=

( )C

thkhk

C

t

D

D +=

221100226.0

3 :

kh kh

( )( ) ( )21

22

khkh

khrw+

=

( )( ) ( )

21

1

hchc

hc

tt

t

+=

8.2

k1h1, k2h2. 3 : , kh kh.

.



178

8.2

8.2

kh. , , , . kh.

.

1, 0, . , ().

3 . . = 0.5 .



179

8.3

8.3

.

. 2- , 20 . . - ( 100 1000) ., 2- .

.

21

2211

hh

hkhkk

+

+=



181

9.1 9.2 9.3



182

9.1

xf

h

w

kf

9.1

. 1000 , : (h), xf, (w).



183

9.1

, ,

2

2

2

f

wD

f

Dxfx

rt

xc

tkt =

=

(

)

(300, )

9.1

. , ( 300). , ( 300).



184

9.1

: -

( ).

300f

f

fDfDCDx

w

k

kwkF ==

4138.1Dxf

CD

D tF

P =

41

4

138.1

ln Dxf

fDfDDxf

D twktd

dp=

9.1

, . .



185

9.1

:

pD

p log-log (quarter slope)

t

Dxf

fDfD

D twk

P 4138.1

=

9.1

. , 4. .

. , 2. .

( ) , - .



186

9.1

9.1

( ) ( ) .

. , . , . .



187

9.1

pwf

mbl

1

4 t

pi

44

215.6tc

k

Fkhx

qBpp

tCDf

iwf

=

4215.6

tCDf

blc

k

Fkhx

qBm

=

4215.6

tCDbl

fc

k

Fkhm

qBx

=

9.1

()

, .



188

9.1

:

:

pD

p log-log

21

DxfD tP = log21log

21log += DxfD tp

9.1

, . , 2. .

( ) , .



189

9.1

()

pwf

ml

1

t

pi tckhx

qB

pptf

iwf

619.0=

tf

lckhx

qBm

619.0=

tl

fckhm

qBx

619.0=

20-30%

9.1

()

, . , 20-30%.



190

9.1

3.14FCD > 3 :

.:

:

:

214.3

1.0

CD

DxF

tf=

214.3

100

CD

DxF

tf=

016.0=fDxt

6.05.0exp5

= CDDx Ft f

9.1

, .



191

9.1

. 95 3/ (600 bbl/d) 300 . 600

.

:

1.2

20 (65.6 ft)

1.5

0.17

1.1* 10-4 atm-1 (7.5* 10-6 psi-1) , FCD = 3

.

9.1

230.13382.00.51547

229.53373.30.39288

228.93365.00.29947

228.43357.10.22827

227.93349.40.17398

227.33342.00.1326

226.93334.90.10107

226.43328.10.07706

225.93321.40.05872

225.53315.00.04477

225.13308.80.03412

224.73302.90.026

224.33297.40.01984

224.03292.30.01511

223.73287.70.01151

223.43283.80.00879

223.23280.40.00668

223.03277.50.0051

222.83275.10.00388

222.73273.10.00296

222.63271.60.00226

222.53270.20.00171

222.43269.30.00131

222.33268.50.00101

222.23265.80

, , psi,

261.33841.4457.32922

260.43828.0348.58411

259.43812.6265.69678

258.23795.2202.51859

256.93776.0154.36313

255.43755.0117.6582

253.93732.889.68106

252.33709.468.35641

250.73685.552.10242

249.13661.139.71332

247.43636.630.27017

245.73612.523.07242

244.13588.717.58618

242.63565.813.40448

241.13543.810.21713

239.73523.07.78766

238.33503.55.93588

237.13485.34.52444

236.03468.53.44861

234.93453.02.62857

233.93438.62.00354

233.03425.31.52713

232.23413.01.164

231.43401.80.88724

230.73391.50.67627



192

9.2

Early time radial flow

Hemiradial f low

Late t ime radial

Linear flow

9.2

,

, . , , , . , . ,

, , , .

:

0.5. khh L(kvkh)0.5

v

h

k

k

L

h5.0



193

9.2

:

==

L

zh

Lc

tkt w

t

hDL 28.1

000355.02

:

25.1000355.0

2 ==

Lc

tkt

t

hDL

9.2

.

. ,

.

, 1.5 2.0.



194

9.2

:

hvw

s

kkLBqm

=195.211 ( )

+

= = 09.3log15.1 2

1

1

w

hvti

crkk

mPPS

()

:

: , , ,

9.2

, , .

. ( ).

: : , , , -



195

9.2

:

hk

Bq

m h

s

195.212=

Sk

k

L

h

SS v

h

g +='

+=

41

sin2

4

v

h

w

w

v

hwg k

k

h

z

r

hLnk

k

L

h

L

rLnS

Sg

- ()

,

9.2

, , , , . .

. .



196

9.2

1.

2. kh

3. p t L

4. ,

, m1

5. -6.

h

v

k

k

9.2

, . -, .

.

, - .



197

9.2

9.2



198

9.3

9.3

- ( + )

- -

,



199

9.3

9.3

: ( ), ( : 1

1, ( , )

- , ,



200

9.3

9.3

:

0.5

, .

, .

w

w

o

o

k

k

5.0



201

9.3

( )wiw SShV

r

=

11

hkBqmw

wws 195.211=

-

+

= 09232.3log1513.12

1

1

wtww

whri

rc

k

m

PPS

hkBqmo

oos 195.212 =

+

= 09232.3log1513.12

2

1

wto

ohri

rc

k

m

PPS

ww

w

o

o

w

w

o

o

r

r

k

kSk

kS 1' ln1

+=

9.3

. , , .

- . , , . -, , .

- - , .



202

9.3

(Abbaszadeh Kamal, 1989)

9.3

(Abbaszadeh Kamal, 1989)

. 2 ( 24 )

, ,

, 4 , 300 ft

, , -



203

10.1

10.2

10.3

10.4 10.5

10.6 :



204

10

10

: , p, t ;

;

;

();

t p, , ,, , .

, , .

, ... , .

Documents

0197804 58D4B Mangazeev p v Pankov m v Kulagina t e Kamartdinov m r Gidrod